Travelling wave equation on one dimension to Gross Pitaeavkii equation is $$ \phi '' +ic\phi'+\phi (1-|\phi|^2)=0\qquad (1) $$ where $c\in (0,\sqrt{2})$ and $ \phi$ is a complex valued function. I am interested on the non-constant T-periodic solutions to equation (1). This post is just to know if someone could give me some notes, papers, books,... where I can read something about and if it is possible to know explicitly all the $T$-periodic solutions. Thanks in advance!

  • $\begingroup$ You can have other frequencies if you adjust the amplitude accordingly. This equation should have solitonic solutions as well. $\endgroup$ Nov 18 '19 at 3:20
  • $\begingroup$ What does if mean adjusting the amplitude? Can You explain that please? $\endgroup$ Nov 18 '19 at 9:03
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    $\begingroup$ If you insert $\phi = Ae^{ikx} $, you get $|A|^2 = 1-k^2 -ck$, so there's a range of solutions for $k$ small enough. Actually, your $Ae^{icx} $ is not a solution for $|A|^2 =1$. $\endgroup$ Nov 18 '19 at 17:39
  • $\begingroup$ Thank you, you are right! By the way, if there a way to prove or disprove that all the periodic solutions to that equation have the form $Ae^{ikx}$ ?? $\endgroup$ Nov 18 '19 at 19:32
  • $\begingroup$ I don't know off-hand. Certainly, there must be more solutions, since there is only one free integration constant in $Ae^{ikx} $, but I don't know whether they're periodic. If there are soliton solutions, maybe one can build soliton-antisoliton chains. I don't know the literature on this, I expect all this is known. $\endgroup$ Nov 18 '19 at 20:51

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